Thanks a lot, but can you maybe also tell me how exactly you can use integration to get drom dv/dt to v ? That would be very nice.

Sure.

The most basic method (which is what I assume you want) is the method of finding an integrating factor.

So what can we multiply the LHS of the equation by to make it the derivative of a product?

Well:

So

(A is the arbitrary constant.)

(If you are more advanced than this, then you might note that the original differential equation is linear and has constant coefficients. So you can solve the homogeneous version of the equation and use variation of parameters to find the particular solution.)